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Descriptive properties of spaces of signed measures. (English) Zbl 1069.54024
Descriptive properties of spaces of signed measures on \(X\) are derived from the respective properties of \(X\). Similar questions for spaces of non-negative measures were answered by P. Holický and O. Kalenda in [Bull. Pol. Acad. Sci., Math. 47, 37–51 (1999; Zbl 0929.54026)]. The results give answers to questions posed by Holický and Kalenda and they say in particular that: If \(Y\) is Borel (Suslin from closed, Suslin from Borel, or co-Suslin from Borel sets) in a Tychonoff space \(X\), then the space \(\mathfrak M_t(Y)\) of signed Radon measures has the same property in \(\mathfrak M_t(X)\subset (C(\beta X)^*,w^*)\). However, there are a compact space \(X\) and an open set \(Y\subset X\) such that \(\mathfrak M_t(Y)\) is not co-Suslin from closed sets in \(\mathfrak M_t(X)\).
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
46E27 Spaces of measures
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